Question: What is the extraneous solution to these equations? $\dfrac{x^2 - 9}{x - 4} = \dfrac{10x - 33}{x - 4}$
Answer: Multiply both sides by $x - 4$ $ \dfrac{x^2 - 9}{x - 4} (x - 4) = \dfrac{10x - 33}{x - 4} (x - 4)$ $ x^2 - 9 = 10x - 33$ Subtract $10x - 33$ from both sides: $ x^2 - 9 - (10x - 33) = 10x - 33 - (10x - 33)$ $ x^2 - 9 - 10x + 33 = 0$ $ x^2 + 24 - 10x = 0$ Factor the expression: $ (x - 6)(x - 4) = 0$ Therefore $x = 6$ or $x = 4$ At $x = 4$ , the denominator of the original expression is 0. Since the expression is undefined at $x = 4$, it is an extraneous solution.